Couplings in the deep infrared limit from M-theory - does one numerical formula deserve the benet of the doubt? Amir Mulic AM Consulting & Publishing PO Box 39, 2018 Løvenstad, Noway; e-mail: [email protected] Abstract In this note, we preliminarily discuss the possibility that the expression −1 3 2 has a physical αem = 4π + π + π interpretation and can even be helpful in model building. If one interprets this expression in terms of the volumes of lp - sized three-cycles on G2 holonomy manifolds and requires that it also comprises eects of the running of the coupling, one can obtain the desired value, but only in a setup which is clearly dierent from the standard model of particle physics (SM). An understanding of the nature of the link between such putative model and SM is needed. Studying this issue could possibly shed some light on existing problems in model building within string theory (ST), particularly the hierarchy problem. Numerological success, which can be achieved if one interprets the formula in terms of volumes of three-cycles on the compactication manifold, as we intend to do here, cannot change the fact that discussion in this note represents merely a heuristic estimate of the feasibility of further research in a certain direction. 1 Discussions regarding the plethora of simple numerical expressions for SM parameters periodically reappear both in printed publications and in blogs, even (or rather especially) in the classical case of −1 The reasons why such attempts are commonly considered to be meaningless and such claims αem: preposterous are well known. A formula extracted directly from compactication geometry could be valid only on very high energies and Renormalization Group (RG) ow to the infrared and symmetry breaking eects should make it unrecognizable in the low energy limit. Therefore, for the sake of argument, it seems appropriate to outline at once why we have chosen to take one of the existing numerical observations as granted. There exists a certain numerical expression which gives a value rather close to −1 in the IR limit, αem which has the form of the sum of powers of π, namely α−1 = 4π3 + π2 + π. This, in turn, suggests P em the familiar 1 1 structure. However, if one wants to have even a remotely realistic toy model, 2 = i 2 e gi one or more (or possibly all) terms should stem from the running of the coupling(s). If this is the case, it is clear that in one-loop expression upper and lower integration limits should be related by an exponential suppression factor of the ∼ exp(−mπn) form , resembling an eect of suppression by instanton ∼ − 1 . Therefore, this picture cannot be a part of SM for at least two reasons. First, exp( g2 ) in such a setup one has the ∼ me scale as an exponentially suppressed ultraviolet (UV) cuto scale, and Higgs condensate is either absent from the picture or somehow identied with UV cuto. Secondly, as there are only three terms, particles other than electrons and positrons are not present in the loops (or all of them are degenerated in mass). As the numerical value is clearly from SM, and should be signicantly inuenced by polarization loops containing particles other than electrons, everything looks like a strange coincidence. Is it possible to turn these problems to possible hints concerning the ST model building? If one intends to proceed, some additional conjectures are inevitable. First, some rationale needes to be provided as to why a crucial SM parameter pops up in one clearly non- SM context. Our most importyant assumptions are the following: SM corresponds to some conguration of branes and other ST objects which either evolved from another conguration - one in which the displayed expression appears natural - or the values of the parameters in the IR limit were somehow forced upon the SM conguration, presumably by some interaction with hidden sector. However, keeping in mind the dependence of the ne structure constant value of the particle spectrum of SM, this would inevitably also mean that the particle content of the SM has been forced upon the system of branes where the gauge elds of SM reside. It would be illuminating to have an understanding of this at least on the toy-model level. Another important idea is that the very presence of suppression by some non-perturbative instanton eect points towards the compactication upon some G2 holonomy manifold, where hierarchies intro- duced in this way are generic (see e.g. the recent review [1] and references therein). Of course, in these developments a hierarchy exists between the Planck and gravitino scales, and, strictly speaking, a proper link between the observation we intend to display and compactication upon G2 manifolds has yet to be determined. Any non-oversimplied calculation of the running of coupling will also take into account the contribu- tions of polarization loops besides the one with the electron and positron; in models beyond SM, there can be even more species of charged particles. Consequently, there is no reason to expect some simple formula here. At the same time, within the framework of ST or M-theory model building, a purely geometrical formula appears to be natural only if connected with the 11-dimensional Planck scale lp. It can be argued 0 that what we actually need are formulas for g and g0 in the familiar SM expression e = p g g and not g02+g2 2 e2 itself. In addition, if one is working with -scale, then, a priori, masses are expected to be of the α = 4π lp Stueckelberg type, and not due to the electroweak Higgs eect. We shall take as granted the expression 4π3 + π2 + π = 137; 036::: (which was probably proposed for the rst time in the early 1970s). Further, when, for example, β-function of Quantum Electrodynam- ics or specic G2 holonomy manifolds and so on are used, they play role of illustrative tools at hand and should not be understood as claims. Here, admittedly, the numerical coincidence substitutes the phenomenological input usually used in preliminary dimensional and numerical estimates. One of the possibilities is that values of the couplings were somehow forced upon the system of branes that actually hosts SM elds, for example, by the SUSY breaking mechanism. Another possibility would be the existence of a protostandard model (ΠSM in further text), which naturally comprises the described expression and belongs to the same universality class as SM. Then, ΠSM would eventually evolve into SM (an idea somehow in spirit of brane realizations of duality cascades). In the latter case an obvious objection can be immediately raised. Formula is very simple, and if it describes some running, it is within some very simple model, with few (likely only one) particle pairs in the polarization loops. At the same time, it has a rather precise value. More charged matter in the abelian case means stronger polarization. Therefore, the concept of some class of models with the same abelian coupling in the IR limit, with the same coupling on unication/Planck scale, seems wrong and requires further modication. An attempt shall be made to address this issue. Needed numerical values can be obtained if one substitutes lp-sized length parameters (e.g. dened as inverse tension of the fundamental M-theory object, namely M2 brane) in the expression for gauge coupling for D6 branes wrapped upon three-cycles on G2 holonomy manifold and in exponential sup- pression factors (of the instantonic type), as well depending upon volumes of the such cycles. Obviously, if presented observation is taken as granted, the problem of gauge coupling values essentially becomes a part of the hierarchy problem, as one starts from a model where the scale of the lightest charged particle equals the nonperturbatively suppressed UV cuto scale. A straightforward estimate of coupling using compactication of D6 branes and E2 instantons upon lp-sized three-cycles on G2 holonomy manifold rather naturally yields the desired value. Finally we shall make some guesses concerning the question how this can be reconciled with the actual content of SM. Eventually one will need both a constructive realization of ΠSM and understanding of its further evolution. 1 One-loop polarization versus the displayed formula The mentioned formula renders a value only slightly outside the accepted interval of measurement results for the inverse of the electromagnetic ne structure constant: −1 3 2 (1) αem = 4π + π + π = 137; 036::: One can observe that using the volume elements on the corresponding homology cycles on the manifold 1 3 S × S , i.e. generators of de Rham cohomology of U(2) v SU(2) × U(1)=Z2, !1;!2;!3 = !1 ^ !2 (not normalized to integer) it is possible to write1: Z 1 "137; 036:::" = (2!3 + !1 + !2) (2) S1×S3 2 1 To our best knowledge expression (2) has never before been published in this form. 3 This may be interesting on its own, but will not be used in this note2. Such expressions seem to contradict the common knowledge about the running of αem. One usually overlooks that, for example, formula (1) can be reconciled with the running of the coupling as such, but not with the running of the coupling in SM. Let us suppose that one tries to match some expression like (1) with the 1-loop formula 1 − 1 Nf Λ (3) 2 2 = 2 ln eR e 6π Ame where Λ stands for some high-energy cuto scale and the denominator is connected to the threshold for creating the pairs of charged particles. It will be useful to express (3) as: 1 1 N Λ − = f ln (4) e2 e2 6π2 <v>p Y R 2 e where < v > is the electroweak Higgs condensate and Ye stands for the Yukawa-coupling responsible for electron mass.